A347926 - OEIS

%I #12 Oct 16 2021 09:02:06

%S 3,7,9,8,3,1,2,1,4,9,2,6,6,1,0,9,0,1,8,2,2,6,1,0,0,5,6,7,2,1,2,2,9,2,

%T 4,4,1,7,6,2,9,1,0,7,2,5,8,6,3,9,1,5,3,3,5,4,8,1,5,6,5,5,5,7,7,6,8,2,

%U 7,1,7,4,5,2,5,2,0,6,3,8,8,9,0,8,4,7,3,7,9,8,0,8,8,7,3,3,4,7,5,8,2,2,8,1,3

%N Decimal expansion of the smallest a such that log(1 + x) <= x^a for all x >= 0.

%C _Fredrik Johansson_ remarks: "The inequality log(1 + x) <= x is used all the time. Putting x^a on the right gives a bound that grows less quickly and which remains easy to manipulate multiplicatively."

%e 0.37983121492661090182261...

%p Digits := 120: Optimization:-Maximize(log(log(1 + x))/log(x), {x>=9})[1]:

%p evalf(%)*10^105: ListTools:-Reverse(convert(floor(%), base, 10));

%t RealDigits[FindMaximum[Log[Log[1 + x]]/Log[x], {x, 7}, WorkingPrecision -> 110][[1]], 10, 105][[1]] (* _Amiram Eldar_, Oct 16 2021 *)

%K nonn,cons

%O 0,1

%A _Peter Luschny_, Oct 16 2021